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Determine the whole number unknowns for the solution set of the given graph of an inequality in two variables.

__ x < __ y

3 answers

To determine the whole number unknowns for the solution set of the inequality \( x < y \), we need to analyze what this inequality represents.

  1. Understanding the Inequality:

    • The inequality \( x < y \) means that the value of \( y \) must be greater than the value of \( x \).
    • This can be considered in a coordinate system (with \( x \) and \( y \) as axes), where the region of interest is above the line \( y = x \).

  2. Finding Whole Number Solutions:

    • We need to find the pairs of whole numbers (non-negative integers) \( (x, y) \) that satisfy the inequality \( x < y \).
    • For whole number solutions, \( x \) can take values from 0 upwards, and for each specific integer value of \( x \), \( y \) must be greater than \( x \).

  3. Example Solutions:

    • If \( x = 0 \), then \( y \) can be \( 1, 2, 3, \ldots \) (i.e., \( y \) can be any whole number greater than \( 0 \)).
    • If \( x = 1 \), then \( y \) can be \( 2, 3, 4, \ldots \) (any whole number greater than \( 1 \)).
    • If \( x = 2 \), then \( y \) can be \( 3, 4, 5, \ldots \) (i.e., any whole number greater than \( 2 \)).
    • This pattern continues indefinitely for all whole numbers.

  4. Formulating the Solution Set:

    • The solution set can be written as pairs \( (x, y) \) such that:
      • \( x \) is a whole number \( \geq 0 \)
      • \( y \) is a whole number \( > x \)

Thus, the whole number unknowns for the solution set of the inequality \( x < y \) consist of infinite pairs of whole numbers \( (x, y) \), where for each whole number \( x \), \( y \) must always be greater than \( x \).

what are the numbers then

To list the specific pairs of whole numbers (non-negative integers) that satisfy the inequality \( x < y \), we can outline some of the possible combinations systematically.

Here’s a breakdown of the values for \( x \) and the corresponding values for \( y \):

  1. For \( x = 0 \):

    • Possible values for \( y \): \( 1, 2, 3, 4, \ldots \)

  2. For \( x = 1 \):

    • Possible values for \( y \): \( 2, 3, 4, 5, \ldots \)

  3. For \( x = 2 \):

    • Possible values for \( y \): \( 3, 4, 5, 6, \ldots \)

  4. For \( x = 3 \):

    • Possible values for \( y \): \( 4, 5, 6, 7, \ldots \)

  5. For \( x = 4 \):

    • Possible values for \( y \): \( 5, 6, 7, 8, \ldots \)

  6. Continuing this pattern:

    • Generally, for any whole number \( x \), \( y \) will be any whole number greater than \( x \).

So, some example pairs of \( (x, y) \) that satisfy \( x < y \) include:

  1. \( (0, 1) \)
  2. \( (0, 2) \)
  3. \( (0, 3) \)
  4. \( (1, 2) \)
  5. \( (1, 3) \)
  6. \( (2, 3) \)
  7. \( (2, 4) \)
  8. \( (3, 4) \)
  9. \( (3, 5) \)
  10. \( (4, 5) \)

And so on...

Summary of Numbers

The full solution set of whole number pairs \( (x, y) \) such that \( x < y \) can be expressed as:

  • For each non-negative integer \( x = 0, 1, 2, 3, \ldots \),
  • The corresponding \( y = 1, 2, 3, 4, \ldots (y > x) \).

In terms of numeric pairs:

\[ {(0,1), (0,2), (0,3), (0,4), \ldots, (1,2), (1,3), (1,4), \ldots, (2,3), (2,4), \ldots} \]

The process gives us an infinite set of pairs where \( y \) is always greater than \( x \).